Final answer:
The question is about using the method of cylindrical shells to compute the volume of a rotated region and finding the second derivative of a trigonometric function with respect to x. The volume and surface area formulas for spheres and cylinders are also mentioned.
Step-by-step explanation:
The question pertains to the method of cylindrical shells and the calculation of volume generated by rotating a given region. The region is bounded by the curves y=3+2x-x² and x+y=3. The method involves integrating over a single variable, which can be either x or y, depending on the simplicity of the function involved. In this context, if one finds the function in terms of y to be complicated due to square root and fractional exponents, x would be a more convenient variable for integration.
To find the second derivative of the function y=4cos(x)+x²sin(x) with respect to x, we first find the first derivative (dy/dx) and then the second derivative (d²y/dx²). Please note that the correct notation for derivatives is dy/dx for the first derivative and d²y/dx² for the second derivative.
The concept of vector addition and its analytical method can also shed light on how to approach complex mathematical problems by breaking them down into more manageable parts. Understanding volume and surface area formulas is crucial for problems involving geometric shapes like cylinders and spheres. Notably, the formula for the volume of a sphere is 4πr³/3, and the formula for its surface area is 4πr².